My 3yo has had an onset of cough for 9 days. Some phlegm. Md says no redness &clear lungs. Seems to happen more at exact moment of eating, not limited to?

See below: He could be still recovering from a viral upper respiratory illness-at his age he is unable to clear & coordinate his post nasal drip mucus moving in the back of his throat. Other possibilities if similar symptoms, yet continuing for many weeks/mos: vasomotor rhinitis-not caused by an infection or allergy. Exact cause is unknown.